Rubin Kazan star Azmoun, 23, has been capped 31 times by his country and has scored 23 goals, and Celtic were credited with an interest in the forward in June last year.
Fulham and Wolves are also said to be keen on the striker but it is believed that Everton’s interest depends on Sam Allardyce’s future as Toffees boss.
The Goodison manager’s future appears uncertain after the club sent a survey to fans asking them to rate Allardyce’s performance out of 10.
In January this year, Rubin Kazan revealed that the Hoops had been interested in Azmoun but had not submitted a bid for the player.
Azmoun, who was born in Gonbad-e Kavus in Iran, joined Rubin Kazan from Sepahan in his homeland and was loaned to Rostov before joining them permanently in 2016.
His performances for Rostov led to interest from clubs including Liverpool, Marseille, Arsenal, Borussia Dortmund and Bayer Leverkusen.
Keen to sign the player permanently, Rostov activated a buy-out clause in Azmoun’s contract, and despite a legal challenge from with Rubin Kazan, the Court of Arbitration for Sport ruled Azmoun was Rostov’s player.
However, the forward returned to Rubin Kazan in the summer of 2017.